Notes on Equivalence and Minimization of Weighted Automata

TL;DR

This paper revisits fundamental results on weighted automata, providing a unified, linear-algebraic perspective on their equivalence and minimization, with a focus on algorithmic aspects and concise proofs.

Contribution

It offers a unified, linear-algebraic presentation of known theorems on weighted automata, emphasizing algorithmic approaches and clarity.

Findings

Re-proves known results on automata equivalence and minimization

Provides a unified linear-algebraic framework for weighted automata

Focuses on elementary, succinct proofs and algorithmic aspects

Abstract

This set of notes re-proves known results on weighted automata (over a field, also known as multiplicity automata). The text offers a unified view on theorems and proofs that have appeared in the literature over decades and were written in different styles and contexts. None of the results reported here are claimed to be new. The content centres around fundamentals of equivalence and minimization, with an emphasis on algorithmic aspects. The presentation is minimalistic. No attempt has been made to motivate the material. Weighted automata are viewed from a linear-algebra angle. As a consequence, the proofs, which are meant to be succinct, but complete and almost self-contained, rely mainly on elementary linear algebra.